linear equations linear inequalities drive on the education highway
TRANSCRIPT
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LinearEquations
Linear Inequalities
Drive on The Education Highway
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Linear Equations and Graphing
1. Parts of a Coordinate Plane
2. Slope
3. Slope-intercept Form of a
Linear Equation
4. Graphing by x- & y-intercepts.
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Parts of a coordinate plane.
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Click on the correct quadrant numbers. Correct answer = applause.
Quadrant
I II III IV
Quadrant
I II III IV
Quadrant
I II III IV
Quadrant
I II III IV
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Click on the correct axis names. Correct answer = clapping.
x-axis
y-axis
x-axis y-axisLessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Click on the point for the origin. Correct answer = clapping.
x
yLessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Click on point (-3, 5). Correct point = applause.
x
yLessonStart
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You chose a line segment instead of a point.
Go back and try again.
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You chose point (5, -3).
Each ordered pair is in the form (x, y) -- it follows the alphabet in its internal order.
You find the x value first, then you find the y value. Where they meet is the point.
Go back and try again.
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Click on the correct ordered pair for the black point.
-5/4
(4, -5)
(-5, 4)
-4/5
x
yLessonStart
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You did not choose an ordered pair.
Go back and try again.
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You chose the ordered pair for the pale green point.
Remember: x comes before y in the alphabet and in an ordered pair.
Go back and try again.
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Return to Main Menu.
Return to Prior Problem.
Continue to Next Lesson.
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Slopes
1. What is a slope?
2. Slope formula
3. Types of slopes
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What is slope?
Slope is the slant of a line.
Slope = rise change in y’s
run change in x’s
Slope is a fraction/integer.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
How to determine the slope when the line goes up.
1. Count the number of units up from the right point to the left point.
1
2
3
45
6
2. Put that number on top of the fraction line.
Slope = 6
3, Count the number of units to the right.
1 2 3 4 5 6 7 8 9
4. Put that number under the fraction line.
9
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
How to determine the slope when the line goes down.
1. Count the number of units down from right point to left point.
-1-2
-3-4
-5-6
2. Put that number on top of fraction line.
Slope = -6
3. Count the units to the right.
1 2 3
4. Put that number under the fraction line.
3
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Determine the slope of the line shown.
-1/3 3/1
-3/1 1/3LessonStart
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The line does not go down.
Go back and try again.
LessonStart
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The line does not rise 3 units,
then run 1 unit to the right.
Go back and try again.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Determine the slope of the line shown.
-2/3 3/2
-3/2 2/3LessonStart
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The line does not go up.
Go back and try again.
LessonStart
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The line does not rise -2 units,
then run 3 units to the right.
Go back and try again.
LessonStart
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Slope Formula:
m = (y1 - y2)(x1 - x2)
where m = slope
and (x1, y1), (x2, y2) are
points on the line.
LessonStart
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Example: Find the slope for a line with points (-3, 4) and (7, -2) on it.1. Assign values as follows:
2. Substitute them into the formula and solve.
m = 4 - (-2) = 6 = -3 -3 - 7 -10 5
(x1, y1) = (-3, 4)
(x2, y2) = (7, -2)
LessonStart
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Find the slope of the line with points (5, 6) and (2, 9) on it.
1. (x1, y1) = (5, 6) (2, 9)
LessonStart
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Find the slope of the line with points (5, 6) and (2, 9) on it.
1. (x1, y1) = (5, 6)
2. (x2, y2) =(5, 6) (2, 9)
LessonStart
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Find the slope of the line with points (5, 6) and (2, 9) on it.
1. (x1, y1) = (5, 6)
2. (x2, y2) = (2, 9)3. m =
6 + 95 + 2
6 - 95 - 2
5 - 26 - 9
5 + 26 + 9
LessonStart
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The slope formula is a case of subtraction
on top and bottom.
Go back and try again.
LessonStart
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You have your x’s and y’s upside down.
Remember:
“Y’s guys are always in the skies!”
Go back and try again.
LessonStart
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You have your x’s and y’s upside down.
You are also adding when you need to subtract.
Go back and try again.
LessonStart
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Find the slope of the line with points (5, 6) and (2, 9) on it.
1. (x1, y1) = (5, 6)
2. (x2, y2) = (2, 9)3. m = 6 - 9 = -3 = -1
5 - 2 3
LessonStart
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Find the slope of the line with points (7, 5) and (3, -4) on it.
m = 5 - 4 = 17 - 3 4
7 - 3 = 45 - (-4) 9
5 - (-4) = 9 3 - 7 -4
5 - (-4) = 9 7 - 3 4
LessonStart
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You have your x’s and y’s upside down.
Remember:
“Y’s guys are always in the skies!”
Go back and try again.
LessonStart
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It is not 5 - 4, it is 5 - (-4).
Go back and try again.
LessonStart
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You must start with the y and x from the first point and end with the y and x from the second
point.
Go back and try again.
LessonStart
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Types of Slopes:
1. Positive and Negative Slopes
2. Special Types of Slopes
3. Determining Types of Slopes by
Looking at Graphs of Lines
4. Determining Types of Slopes
Algebraically.LessonStart
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Positive and Negative Slopes.
Type Graph AlgebraPositive Up left to right. Positive
Fraction
Negative Down left to right. Negative
FractionLessonStart
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2 Special Types of Slopes
Type Graphs AlgebraZero Horizontal Line 0/a, a 0
UndefinedVertical Line a/0
No Slope
LessonStart
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Determining Types of Slopes by
Looking at Graphs of Lines
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Is the slope of the line positive, negative, zero, or undefined?
-0 +LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Is the slope of the line positive, negative, zero, or undefined?
-0+LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Is the slope of the line positive, negative, zero, or undefined?
-0 +LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Is the slope of the line positive, negative, zero, or undefined?
-0 +LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Click on the line with the negative slope.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Click on the line with the zero slope.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Click on the line with the no slope.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Click on the line with the positive slope.
LessonStart
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Determining Types of Slopes
Algebraically.
LessonStart
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Is the slope of the line with the 2 points listed positive, negative, zero, or undefined?
Points (-3, 5) and (-9, -4)
+ - 0
LessonStart
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Is the slope of the line with the 2 points listed positive, negative, zero, or undefined?
Points (3, 5) and (3, -4)
+ - 0
LessonStart
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Is the slope of the line with the 2 points listed positive, negative, zero, or undefined?
Points (-3, -5) and (-9, -4)
+ - 0
LessonStart
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Is the slope of the line with the 2 points listed positive, negative, zero, or undefined?
Points (-3, -4) and (-9, -4)
+ - 0
LessonStart
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Return to Main Menu.
Return to Prior Problem.
Continue to Next Lesson.
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Slope-intercept Form
of a Linear Equation
1. Slope-intercept equation
2. Graphing by slope-intercept
3. Writing slope-intercept equations
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Slope-intercept Form:
y = mx + b
where m = slope
and b = y-intercept.
LessonStart
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Example:
y = -1/2x + 4
-1/2 = m = slope
4 = b = y-intercept
LessonStart
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Graphing by Slope-intercept
1. Graphing lines with
positive/negative slopes.
2. Graphing lines with zero or
undefined/no slopes.
LessonStart
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The Slope-intercept SongYou make the last number first.
It’s either up or down.
Make the slope in 2 numbers,
Or you look like a clown.
Top one’s up or down,
And the bottom’s always right.
You’d better do it well,
Or you’ll get a fright. (Tune: “Hokey-Pokey”)
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Graph y = -1/2x + 4
1. Last number is 4, so go up 4 on the y-axis from the origin and plot a point.
1
2
3
2. Slope is already 2 numbers. Top one is -1, so go down 1 from the point you just plotted.
3. The bottom number is 2, so go 2 units to the right and plot a point.
4. Draw a line through the 2 points you plotted.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Graph y = 2x - 3
1. Last number is -3, so go down 3 units from the origin and plot a point.
1
2
2. The slope is only 1 number so put a 1 under the 2.
3. Go up 2 from the point you just plotted.
4. Go 1 unit to the right and plot a point.
5. Draw a line through the 2 points you plotted.
1
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Click on the graph for y = 2/3x - 2
LessonStart
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The slope is not negative.
Go back and try again.
LessonStart
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You graphed the last number on the x-axis instead of the y-axis.
Go back and try again.
LessonStart
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Top number is 2, and the bottom is 3,
so you do not go up 3 and over 2.
Go back and try again.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Click on the graph for y = -4x + 3
LessonStart
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The slope is not positive.
Go back and try again.
LessonStart
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The -4 is not the y-intercept,
nor is the 3 the x-intercept.
Go back and try again.
LessonStart
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The -4 is the slope, not the x-intercept.
Go back and try again.
LessonStart
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Two Special Graphs:Line with a zero slope
And
Line with an undefined slope.Lesson
Start
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Line with a zero slope:
y = # (no x)
graphs as a horizontal line.
“Why, o y, do I look upon the horizon?”
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Graph the equation y = 2.
LessonStart
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Line with an undefined/no slope:
x = # (no y)
graphs as a vertical line.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Graph the equation x = -4.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Click on the graph for x = 3.
LessonStart
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You chose the graph for x = -3.
Go back and try again.
LessonStart
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The x = # (no y) line does not graph
as a horizontal line.
Go back and try again.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Click on the graph for y = -3½.
LessonStart
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The y = # (no x) line does not graph
as a vertical line.
Go back and try again.
LessonStart
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You chose the graph for y = 3½.
Go back and try again.
LessonStart
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Writing Slope-intercept Equations:
1. When given a slope and the
y-intercept.
2. When given a slope and one point
on the line.
3. When given 2 points on the line.
m = ¾, b = -1
m = -¼, (8, 3)
(3, 7), (5, 12)
LessonStart
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Writing a slope-intercept equation when given a slope and the y-
intercept.
Substitute the slope and the y-intercept
for the m and b in the equation.
Example: m = ¾, b = -1
y = mx + b Slope-int. equation
y = ¾x - 1 The new equation
LessonStart
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1. Click on the correct equation for a line with
slope = 5/3 and y-intercept = 2.y = 5/3x + 2y = 2x + 5/35/3y = 2x y = -5/3x + 2
2. Click on the correct slope and y-intercept
pair for y = 7x - 5.
m = 7, b = -5 m = -5, b = 7 m = -7, b = 5 m = 1/7, b = -5
LessonStart
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Writing a slope-intercept equation when given a slope and a point on the
line.
1. Substitute the x, y, and m in the
slope-intercept form.
2. Solve for b.
3. Substitute the slope and the b in a
clean slope-intercept form.LessonStart
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Example: Write the equation of the line with
slope = -¼ and point (8, 3). y = mx + b
3 = -¼(8) + b
1. Substitute the slope, x, and y in the equation.
3 = -2 + b
+2 +2
5 = b
2. Solve for b.
y = -¼x + 5
3. Substitute the slope and b in the equation.
LessonStart
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1. Click on the correct substitution for a line
with slope = 1/3 and point (5, 9).
9 = 1/3(5) + b 5 = 1/3(9) + b9 = 1/3x + 59y = 5x + 1/3
2. Click on the correct equation for a line
with slope = -2/3 and point (-6, 4).
y = -2/3xY = -2/3x + 4y = -2/3y = -2/3x - 6
LessonStart
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Writing a slope-intercept equation for a line with 2 points given:
1. Find the slope of the line.
2. Use that slope and the first point to
find the y-intercept.
3. Substitute the slope and the
y-intercept into the equation.
LessonStart
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Example: Write and equation for a
line with points (3, 7) & (5, 12).
1. Find the slope of the line.
m = (y1 - y2)
(x1 - x2)
m = 7 - 12 = -5 = 5
3 - 5 -2 2
Continued on next screen.LessonStart
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Write and equation for a line with
points (3, 7) & (5, 12).
m = 5/2 y = mx + b
2. Use the slope and
the first point to
solve for the
y-intercept.
7 = (5/2)(3) + b
2(7) = 2(15/2) + 2b
14 = 15 + 2b
-15 -15
-1 = 2b -1/2 = b
2 2Continued on next screen.Lesson
Start
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Write and equation for a line with
points (3, 7) & (5, 12).
m = 5/2, b = -1/2
3. Substitute the slope and the y-intercept for the m and the b in the equation.
y = mx + b
y = 5/2x - 1/2
LessonStart
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1. Click on the slope for a line with points
(-2, 8) and (7, -5).13-9
3-9
-913
139
2. Click on the y-intercept for a line with
points (-2, 8) and (7, -5).469
869
-58613
LessonStart
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3. Click on the correct equation for a line
with points (3, 7) and (4, 8).
y = x + 4
y = 3x + 7
y = -x + 4
y = 3/4x + 8
LessonStart
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Return to Main Menu.
Return to Prior Problem.
Continue to Next Lesson.
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Graphing by x- and y-intercepts.
X-intercept: where the line crosses
the x-axis.
Y-intercept: where the line crosses
the y-axis.
x
y
x-intercept
y-intercept
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How to graph by x- & y-intercepts:
1. Cover the x term with your index finger and solve the resulting equation for y.
2. Go up or down on the y-axis from the origin that many units and plot a point.
3. Cover the y term with your index finger and solve the resulting equation for x.
4. Go left or right on the x-axis from the origin that many units and plot a point.
5. Draw a line through your points.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
Graph the line for 3x + 2y = 6.
1. Cover the x term and solve for y.
3x + 2y = 6.
y = 3
2. Go up 3 units on the y-axis.
1
2
3. Cover the y term and solve for x.
3x + 2y = 6.
x = 2
4. Go right 2 units on the x-axis.
1
5. Draw a line through the points plotted.
LessonStart
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1. Click on the correct intercepts for
3x - 4y = 24.
x-int: 8y-int: -6
x-int: -6y-int: 8
x-int: 8y-int: 6
x-int: 6y-int: 8
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
2. Click on the graph of 3x - 6y = 12.
LessonStart
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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
5
4
3
2
1
-1
-2
-3
-4
x
y
3. Click on the correct equation for the line shown.
-6x - 9y= -36
-6x - 9y= -36
-9y - 6x= -36
-9y - 6x= -36
4x + 6y = 36
4x + 6y = 36
6y + 4x= 36
6y + 4x= 36
LessonStart
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Return to Main Menu.
Return to Prior Problem.
Continue to Next Lesson.
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Graphing Linear Inequalities
Type of
Line
Where to
Shade
Solving Inequalities
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How to Determine
the Type of Line to Draw
Inequality Symbol
Type of Line
> or < Dotted Line
> or < Solid Line
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Choose the type of line for the inequality
given.1. y > 3x - 2
a. Solid b. Dotted
2. y > ¼x - 5
a. Solid b. Dotted
LessonStart
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Choose the inequality symbol for the line shown.
< or >
< or >
LessonStart
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Choose the inequality symbol for the line shown.
< or >
< or >
LessonStart
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For Positive, Negative, & Zero Slopes
For Undefined
or No Slopes
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Where to Shade for Positive, Negative, and Zero Slopes:
The inequality must be in
y mx + b
format.
can be:
>, >, <, or <.
LessonStart
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If the inequality is:
Shade
y > mx + bor
y > mx + bAbove the
line
y < mx + bor
y < mx + bBelow the
lineLessonStart
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x
y
Graph y > x - 2.1. Graph the line
y = x - 2.
2. Since y >, shade above the line.
LessonStart
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x
y
Graph y < x - 2.1. Graph the line
y = x - 2.
2. Since y <, shade below the line.
LessonStart
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Do you do anything
different when the line is
dotted rather than solid?
LessonStart
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LessonStart
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x
y
Graph y > x - 2.
2. Since y >, shade above the line.
1. Graph the line
y = x - 2, but make the line dotted.
LessonStart
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x
y
Graph y < x - 2.1. Graph the line
y = x - 2, but make the line dotted.
2. Since y <, shade below the line.
LessonStart
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x
y
Graph y > -½x + 3Type of
line:
Solid
Dotted
LessonStart
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x
y
Graph y > -½x + 3Type of
line:
Solid
Dotted
Shade ___ the line.
Above Below
LessonStart
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x
y
Graph y > -½x + 3Type of
line:
Solid
Dotted
Shade ___ the line.
Above Below
LessonStart
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x
y
Choose the correct inequality for the graph shown.
y < 1/3 x + 2
y < 1/3 x + 2
y > 1/3 x + 2
y > 1/3 x + 2
LessonStart
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Where to Shade for Undefined or No Slopes:
The inequality must be in
x # (no y)
format.
can be:
>, >, <, or <.
LessonStart
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If the inequality is:
ShadeTo the
x > #or
x > #Right of the
line
x < #or
x < #Left of the line
LessonStart
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x
y
Graph x > -21. Draw a dotted vertical line at x = -2.
2. Shade to the right of the line.
LessonStart
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x
y
Graph x < -2.1. Graph the line
X = -2.
2. Shade to the left of the line.
LessonStart
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x
y
Graph x > 3.Choose type of
line.
Solid
Dotted
LessonStart
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x
y
Graph x > 3.Choose type of
line.
Solid
Choose where to shade.
Left Right
LessonStart
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x
y
Graph x > 3.Choose type of
line.
Solid
Choose where to shade.
Right
LessonStart
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Solving Inequalities
You use the same algebraic methods as solving equations except when you multiply/divide both sides by the same negative number.
In that case, you switch the direction of the inequality symbol.
LessonStart
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Solve -3x - 2y < 12.
-3x - 2y < 12+3x +3x
-2y < 3x + 12-2 -2 -2 y < -3/2 x - 6>
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Choose the correct inequality.
1. 2x + 5y > -10
y > -2/5 x - 2y < -2/5 x - 2
y > 2/5 x + 2 y < 2/5 x + 2
2. 3x - 2y > 10y > -2/3 x - 5 y < -2/3 x - 5
y > 2/3 x - 5y < 2/3 x - 5
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